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Mathematics is the study of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. It evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

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The continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:

There is no set whose size is strictly between that of the integers and that of the real numbers.

Or mathematically speaking, noting that the cardinality for the integers $|\mathbb{Z}|$ is $\aleph_0$ ("aleph-null") and the cardinality of the real numbers $|\mathbb{R}|$ is $2^{\aleph_0}$ , the continuum hypothesis says:

$\nexists \mathbb{A}: \aleph_0 < |\mathbb{A}| < 2^{\aleph_0}.$

This is equivalent to:

$2^{\aleph_0} = \aleph_1$

The real numbers have also been called the continuum, hence the name.

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Credit: Jtico

In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:

In a projective space, two triangles are in perspective axially if and only if they are in perspective centrally.

The above picture illustrates Desargues' theorem. Another important feature of projective geometry noticable in the picture is all lines meet at exactly one point (ie there are no parallel lines).

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Did you know...

...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
...that a sphere can be cut up and reassembled into two spheres the same size as the original (Banach-Tarski paradox)?
...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.
...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
...that every natural number can be written as the sum of four squares?

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Topics in mathematics

General	Foundations	Number theory	Discrete mathematics
Mathematicians History of mathematics Philosophy of mathematics Mathematical notation Mathematical beauty Mathematics education Areas of mathematics	Foundations of mathematics Mathematical logic Set theory (portal) Naive set theory Axiomatic set theory Proof theory Model theory Recursion theory Category theory (portal) Topos theory Gödel's incompleteness theorems	Number theory (portal) Algebraic number theory Analytic number theory Arithmetic Numbers Natural numbers Prime numbers Rational numbers Algebraic numbers Fundamental theorem of arithmetic	Discrete mathematics (portal) Combinatorics Graph theory Game theory Discrete geometry Theoretical computer science Computability theory Complexity theory Information theory
Analysis	Algebra	Geometry and topology	Applied mathematics
Analysis (portal) Calculus Vector calculus Differential equations Calculus of variations Real analysis Complex analysis Functional analysis Measure theory Harmonic analysis Special functions Fundamental theorem of calculus	Algebra (portal) Elementary algebra Abstract algebra Group theory Ring theory Field theory Commutative algebra Linear algebra Matrix theory Multilinear algebra Universal algebra Order theory Fundamental theorem of algebra	Geometry (portal) Topology (portal) General topology Algebraic topology Geometric topology Euclidean geometry Non-Euclidean geometry Affine geometry Projective geometry Differential geometry Riemannian geometry Lie groups Algebraic geometry Trigonometry	Applied mathematics Numerical analysis Mathematical biology Optimization Dynamical systems Chaos theory Financial mathematics Cryptography (portal) Mathematical physics Classical mechanics Probability Statistics Stochastic processes Operations research

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